第 19900 題：Fermat小定理

原題與已知條件

exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={"p":97,"a":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={"p":97,"a":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-fermat-little-theorem

逐步推導

source operation=flt-prime-coprime;original snapshot and mathematics retained

explicit ordered source output fields=[p, normalized_base, power_p_minus_1, congruence_certificate, boundary]

ACTUAL integer modulus=97;complete trial range2..floor_sqrt=9

checked rows[divisor,quotient,remainder]=[[2, 48, 1], [3, 32, 1], [4, 24, 1], [5, 19, 2], [6, 16, 1], [7, 13, 6], [8, 12, 1], [9, 10, 7]];first exact factor=null;prime=是

original signed base=26;positive modulus=97;use nonnegative gcd inputs=[26, 97]

ALL Euclidean rows[left,right,q,r]=[[26, 97, 0, 26], [97, 26, 3, 19], [26, 19, 1, 7], [19, 7, 2, 5], [7, 5, 1, 2], [5, 2, 2, 1], [2, 1, 2, 0]];terminal nonnegative gcd=1;coprime=是

Fermat original signed base exact quotient := div(26, 97)

Fermat original signed base exact quotient = 26/97

Fermat original signed base floor quotient multiple := mul(0, 97)

Fermat original signed base floor quotient multiple = 0

Fermat original signed base normalized remainder := sub(26, 0)

Fermat original signed base normalized remainder = 26

Fermat original signed base:signed integer value=26;positive modulus=97;exact quotient=26/97;floor q=0

value=q*modulus+remainder;normalized remainder=26 in[0,97)

Fermat original modular exponent:original normalized base=26;original exponent=96;modulus=97

initialize accumulator1,current_base=base,remaining=exponent;invariant accumulator*current_base^remaining congruent to original power

Fermat original modular exponent bit0 current base square := mul(26, 26)

Fermat original modular exponent bit0 current base square = 676

Fermat original modular exponent bit0 squared base reduction exact quotient := div(676, 97)

Fermat original modular exponent bit0 squared base reduction exact quotient = 676/97

Fermat original modular exponent bit0 squared base reduction floor quotient multiple := mul(6, 97)

Fermat original modular exponent bit0 squared base reduction floor quotient multiple = 582

Fermat original modular exponent bit0 squared base reduction normalized remainder := sub(676, 582)

Fermat original modular exponent bit0 squared base reduction normalized remainder = 94

Fermat original modular exponent bit0 squared base reduction:signed integer value=676;positive modulus=97;exact quotient=676/97;floor q=6

value=q*modulus+remainder;normalized remainder=94 in[0,97)

Fermat original modular exponent:step=0;previous[accumulator,base,exponent]=[1, 26, 96];low bit=0

next[accumulator,base,exponent]=[1, 94, 48];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit1 current base square := mul(94, 94)

Fermat original modular exponent bit1 current base square = 8836

Fermat original modular exponent bit1 squared base reduction exact quotient := div(8836, 97)

Fermat original modular exponent bit1 squared base reduction exact quotient = 8836/97

Fermat original modular exponent bit1 squared base reduction floor quotient multiple := mul(91, 97)

Fermat original modular exponent bit1 squared base reduction floor quotient multiple = 8827

Fermat original modular exponent bit1 squared base reduction normalized remainder := sub(8836, 8827)

Fermat original modular exponent bit1 squared base reduction normalized remainder = 9

Fermat original modular exponent bit1 squared base reduction:signed integer value=8836;positive modulus=97;exact quotient=8836/97;floor q=91

value=q*modulus+remainder;normalized remainder=9 in[0,97)

Fermat original modular exponent:step=1;previous[accumulator,base,exponent]=[1, 94, 48];low bit=0

next[accumulator,base,exponent]=[1, 9, 24];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit2 current base square := mul(9, 9)

Fermat original modular exponent bit2 current base square = 81

Fermat original modular exponent bit2 squared base reduction exact quotient := div(81, 97)

Fermat original modular exponent bit2 squared base reduction exact quotient = 81/97

Fermat original modular exponent bit2 squared base reduction floor quotient multiple := mul(0, 97)

Fermat original modular exponent bit2 squared base reduction floor quotient multiple = 0

Fermat original modular exponent bit2 squared base reduction normalized remainder := sub(81, 0)

Fermat original modular exponent bit2 squared base reduction normalized remainder = 81

Fermat original modular exponent bit2 squared base reduction:signed integer value=81;positive modulus=97;exact quotient=81/97;floor q=0

value=q*modulus+remainder;normalized remainder=81 in[0,97)

Fermat original modular exponent:step=2;previous[accumulator,base,exponent]=[1, 9, 24];low bit=0

next[accumulator,base,exponent]=[1, 81, 12];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit3 current base square := mul(81, 81)

Fermat original modular exponent bit3 current base square = 6561

Fermat original modular exponent bit3 squared base reduction exact quotient := div(6561, 97)

Fermat original modular exponent bit3 squared base reduction exact quotient = 6561/97

Fermat original modular exponent bit3 squared base reduction floor quotient multiple := mul(67, 97)

Fermat original modular exponent bit3 squared base reduction floor quotient multiple = 6499

Fermat original modular exponent bit3 squared base reduction normalized remainder := sub(6561, 6499)

Fermat original modular exponent bit3 squared base reduction normalized remainder = 62

Fermat original modular exponent bit3 squared base reduction:signed integer value=6561;positive modulus=97;exact quotient=6561/97;floor q=67

value=q*modulus+remainder;normalized remainder=62 in[0,97)

Fermat original modular exponent:step=3;previous[accumulator,base,exponent]=[1, 81, 12];low bit=0

next[accumulator,base,exponent]=[1, 62, 6];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit4 current base square := mul(62, 62)

Fermat original modular exponent bit4 current base square = 3844

Fermat original modular exponent bit4 squared base reduction exact quotient := div(3844, 97)

Fermat original modular exponent bit4 squared base reduction exact quotient = 3844/97

Fermat original modular exponent bit4 squared base reduction floor quotient multiple := mul(39, 97)

Fermat original modular exponent bit4 squared base reduction floor quotient multiple = 3783

Fermat original modular exponent bit4 squared base reduction normalized remainder := sub(3844, 3783)

Fermat original modular exponent bit4 squared base reduction normalized remainder = 61

Fermat original modular exponent bit4 squared base reduction:signed integer value=3844;positive modulus=97;exact quotient=3844/97;floor q=39

value=q*modulus+remainder;normalized remainder=61 in[0,97)

Fermat original modular exponent:step=4;previous[accumulator,base,exponent]=[1, 62, 6];low bit=0

next[accumulator,base,exponent]=[1, 61, 3];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit5 accumulator product := mul(1, 61)

Fermat original modular exponent bit5 accumulator product = 61

Fermat original modular exponent bit5 accumulator reduction exact quotient := div(61, 97)

Fermat original modular exponent bit5 accumulator reduction exact quotient = 61/97

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple := mul(0, 97)

Fermat original modular exponent bit5 accumulator reduction floor quotient multiple = 0

Fermat original modular exponent bit5 accumulator reduction normalized remainder := sub(61, 0)

Fermat original modular exponent bit5 accumulator reduction normalized remainder = 61

Fermat original modular exponent bit5 accumulator reduction:signed integer value=61;positive modulus=97;exact quotient=61/97;floor q=0

value=q*modulus+remainder;normalized remainder=61 in[0,97)

Fermat original modular exponent bit5 current base square := mul(61, 61)

Fermat original modular exponent bit5 current base square = 3721

Fermat original modular exponent bit5 squared base reduction exact quotient := div(3721, 97)

Fermat original modular exponent bit5 squared base reduction exact quotient = 3721/97

Fermat original modular exponent bit5 squared base reduction floor quotient multiple := mul(38, 97)

Fermat original modular exponent bit5 squared base reduction floor quotient multiple = 3686

Fermat original modular exponent bit5 squared base reduction normalized remainder := sub(3721, 3686)

Fermat original modular exponent bit5 squared base reduction normalized remainder = 35

Fermat original modular exponent bit5 squared base reduction:signed integer value=3721;positive modulus=97;exact quotient=3721/97;floor q=38

value=q*modulus+remainder;normalized remainder=35 in[0,97)

Fermat original modular exponent:step=5;previous[accumulator,base,exponent]=[1, 61, 3];low bit=1

next[accumulator,base,exponent]=[61, 35, 1];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent bit6 accumulator product := mul(61, 35)

Fermat original modular exponent bit6 accumulator product = 2135

Fermat original modular exponent bit6 accumulator reduction exact quotient := div(2135, 97)

Fermat original modular exponent bit6 accumulator reduction exact quotient = 2135/97

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple := mul(22, 97)

Fermat original modular exponent bit6 accumulator reduction floor quotient multiple = 2134

Fermat original modular exponent bit6 accumulator reduction normalized remainder := sub(2135, 2134)

Fermat original modular exponent bit6 accumulator reduction normalized remainder = 1

Fermat original modular exponent bit6 accumulator reduction:signed integer value=2135;positive modulus=97;exact quotient=2135/97;floor q=22

value=q*modulus+remainder;normalized remainder=1 in[0,97)

Fermat original modular exponent bit6 current base square := mul(35, 35)

Fermat original modular exponent bit6 current base square = 1225

Fermat original modular exponent bit6 squared base reduction exact quotient := div(1225, 97)

Fermat original modular exponent bit6 squared base reduction exact quotient = 1225/97

Fermat original modular exponent bit6 squared base reduction floor quotient multiple := mul(12, 97)

Fermat original modular exponent bit6 squared base reduction floor quotient multiple = 1164

Fermat original modular exponent bit6 squared base reduction normalized remainder := sub(1225, 1164)

Fermat original modular exponent bit6 squared base reduction normalized remainder = 61

Fermat original modular exponent bit6 squared base reduction:signed integer value=1225;positive modulus=97;exact quotient=1225/97;floor q=12

value=q*modulus+remainder;normalized remainder=61 in[0,97)

Fermat original modular exponent:step=6;previous[accumulator,base,exponent]=[61, 35, 1];low bit=1

next[accumulator,base,exponent]=[1, 61, 0];split exponent=2*q+bit preserves original-power invariant

Fermat original modular exponent:remaining exponent0;final normalized accumulator=1

EXACT original base^96 modulo97=1

Fermat actual unit inverse product := mul(26, 56)

Fermat actual unit inverse product = 1456

Fermat actual inverse identity exact quotient := div(1456, 97)

Fermat actual inverse identity exact quotient = 1456/97

Fermat actual inverse identity floor quotient multiple := mul(15, 97)

Fermat actual inverse identity floor quotient multiple = 1455

Fermat actual inverse identity normalized remainder := sub(1456, 1455)

Fermat actual inverse identity normalized remainder = 1

Fermat actual inverse identity:signed integer value=1456;positive modulus=97;exact quotient=1456/97;floor q=15

value=q*modulus+remainder;normalized remainder=1 in[0,97)

VERIFIED prime p=97;nonzero original base residue=26;ACTUAL inverse witness=56

multiplication r->a*r mod p permutes ALL nonzero residues1..p-1:inverse gives injectivity and finite self-injection gives surjectivity

same verified prime p=97;ALL nonzero residues permuted by multiplication by a=26

product over r=1..p-1 gives a^(p-1)*(p-1)! congruent to (p-1)!;EVERY factor is a prime-field unit,so factorial is a UNIT;cancel ONLY this nonzero unit to obtain a^(p-1) congruent1

逐步反向核對

• S068 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S067 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S066 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S065 · premise_and_result_audit — 以精確有理數重算 Fermat actual inverse identity normalized remainder，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Fermat actual inverse identity floor quotient multiple，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Fermat actual inverse identity exact quotient，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Fermat actual unit inverse product，不以小數近似替代。
• S061 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S058 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction normalized remainder，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction floor quotient multiple，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 squared base reduction exact quotient，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 current base square，不以小數近似替代。
• S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S053 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 accumulator reduction normalized remainder，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 accumulator reduction floor quotient multiple，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 accumulator reduction exact quotient，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit6 accumulator product，不以小數近似替代。
• S049 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 squared base reduction normalized remainder，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 squared base reduction floor quotient multiple，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 squared base reduction exact quotient，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 current base square，不以小數近似替代。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 accumulator reduction normalized remainder，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 accumulator reduction floor quotient multiple，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 accumulator reduction exact quotient，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit5 accumulator product，不以小數近似替代。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 squared base reduction normalized remainder，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 squared base reduction floor quotient multiple，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 squared base reduction exact quotient，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit4 current base square，不以小數近似替代。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction normalized remainder，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction floor quotient multiple，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 squared base reduction exact quotient，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit3 current base square，不以小數近似替代。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S024 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction normalized remainder，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction floor quotient multiple，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 squared base reduction exact quotient，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit2 current base square，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base reduction normalized remainder，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base reduction floor quotient multiple，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 squared base reduction exact quotient，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit1 current base square，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction normalized remainder，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction floor quotient multiple，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 squared base reduction exact quotient，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Fermat original modular exponent bit0 current base square，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base normalized remainder，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base floor quotient multiple，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Fermat original signed base exact quotient，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(97, 26, 1, 是, prime_coprime_form)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。